Symmetry (physics)

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Symmetry_(physics)

First Brillouin zone of FCC lattice showing symmetry labels

In physics, a symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation.

A family of particular transformations may be continuous (such as rotation of a circle) or discrete (e.g., reflection of a bilaterally symmetric figure, or rotation of a regular polygon). Continuous and discrete transformations give rise to corresponding types of symmetries. Continuous symmetries can be described by Lie groups while discrete symmetries are described by finite groups (see Symmetry group).

These two concepts, Lie and finite groups, are the foundation for the fundamental theories of modern physics. Symmetries are frequently amenable to mathematical formulations such as group representations and can, in addition, be exploited to simplify many problems.

Arguably the most important example of a symmetry in physics is that the speed of light has the same value in all frames of reference, which is described in special relativity by a group of transformations of the spacetime known as the Poincaré group. Another important example is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations, which is an important idea in general relativity.

As a kind of invariance

Invariance is specified mathematically by transformations that leave some property (e.g. quantity) unchanged. This idea can apply to basic real-world observations. For example, temperature may be homogeneous throughout a room. Since the temperature does not depend on the position of an observer within the room, we say that the temperature is invariant under a shift in an observer's position within the room.

Similarly, a uniform sphere rotated about its center will appear exactly as it did before the rotation. The sphere is said to exhibit spherical symmetry. A rotation about any axis of the sphere will preserve how the sphere "looks".

Invariance in force

The above ideas lead to the useful idea of invariance when discussing observed physical symmetry; this can be applied to symmetries in forces as well.

For example, an electric field due to an electrically charged wire of infinite length is said to exhibit cylindrical symmetry, because the electric field strength at a given distance r from the wire will have the same magnitude at each point on the surface of a cylinder (whose axis is the wire) with radius r. Rotating the wire about its own axis does not change its position or charge density, hence it will preserve the field. The field strength at a rotated position is the same. This is not true in general for an arbitrary system of charges.

In Newton's theory of mechanics, given two bodies, each with mass m, starting at the origin and moving along the x-axis in opposite directions, one with speed v₁ and the other with speed v₂ the total kinetic energy of the system (as calculated from an observer at the origin) is 1/2m(v₁² + v₂²) and remains the same if the velocities are interchanged. The total kinetic energy is preserved under a reflection in the y-axis.

The last example above illustrates another way of expressing symmetries, namely through the equations that describe some aspect of the physical system. The above example shows that the total kinetic energy will be the same if v₁ and v₂ are interchanged.

Local and global

Symmetries may be broadly classified as global or local. A global symmetry is one that keeps a property invariant for a transformation that is applied simultaneously at all points of spacetime, whereas a local symmetry is one that keeps a property invariant when a possibly different symmetry transformation is applied at each point of spacetime; specifically a local symmetry transformation is parameterised by the spacetime co-ordinates, whereas a global symmetry is not. This implies that a global symmetry is also a local symmetry. Local symmetries play an important role in physics as they form the basis for gauge theories.

Continuous

The two examples of rotational symmetry described above – spherical and cylindrical – are each instances of continuous symmetry. These are characterised by invariance following a continuous change in the geometry of the system. For example, the wire may be rotated through any angle about its axis and the field strength will be the same on a given cylinder. Mathematically, continuous symmetries are described by transformations that change continuously as a function of their parameterization. An important subclass of continuous symmetries in physics are spacetime symmetries.

Spacetime

Main article: Spacetime symmetries

Lie groups

Continuous spacetime symmetries are symmetries involving transformations of space and time. These may be further classified as spatial symmetries, involving only the spatial geometry associated with a physical system; temporal symmetries, involving only changes in time; or spatio-temporal symmetries, involving changes in both space and time.

• Time translation: A physical system may have the same features over a certain interval of time Δt; this is expressed mathematically as invariance under the transformation t → t + a for any real parameters t and t + a in the interval. For example, in classical mechanics, a particle solely acted upon by gravity will have gravitational potential energy mgh when suspended from a height h above the Earth's surface. Assuming no change in the height of the particle, this will be the total gravitational potential energy of the particle at all times. In other words, by considering the state of the particle at some time t₀ and also at t₀ + a, the particle's total gravitational potential energy will be preserved.
• Spatial translation: These spatial symmetries are represented by transformations of the form r→ → r→ + a→ and describe those situations where a property of the system does not change with a continuous change in location. For example, the temperature in a room may be independent of where the thermometer is located in the room.
• Spatial rotation: These spatial symmetries are classified as proper rotations and improper rotations. The former are just the 'ordinary' rotations; mathematically, they are represented by square matrices with unit determinant. The latter are represented by square matrices with determinant −1 and consist of a proper rotation combined with a spatial reflection (inversion). For example, a sphere has proper rotational symmetry. Other types of spatial rotations are described in the article Rotation symmetry.
• Poincaré transformations: These are spatio-temporal symmetries which preserve distances in Minkowski spacetime, i.e. they are isometries of Minkowski space. They are studied primarily in special relativity. Those isometries that leave the origin fixed are called Lorentz transformations and give rise to the symmetry known as Lorentz covariance.
• Projective symmetries: These are spatio-temporal symmetries which preserve the geodesic structure of spacetime. They may be defined on any smooth manifold, but find many applications in the study of exact solutions in general relativity.
• Inversion transformations: These are spatio-temporal symmetries which generalise Poincaré transformations to include other conformal one-to-one transformations on the space-time coordinates. Lengths are not invariant under inversion transformations but there is a cross-ratio on four points that is invariant.

Mathematically, spacetime symmetries are usually described by smooth vector fields on a smooth manifold. The underlying local diffeomorphisms associated with the vector fields correspond more directly to the physical symmetries, but the vector fields themselves are more often used when classifying the symmetries of the physical system.

Some of the most important vector fields are Killing vector fields which are those spacetime symmetries that preserve the underlying metric structure of a manifold. In rough terms, Killing vector fields preserve the distance between any two points of the manifold and often go by the name of isometries.

Discrete

Main article: Discrete symmetry

A discrete symmetry is a symmetry that describes non-continuous changes in a system. For example, a square possesses discrete rotational symmetry, as only rotations by multiples of right angles will preserve the square's original appearance. Discrete symmetries sometimes involve some type of 'swapping', these swaps usually being called reflections or interchanges.

• Time reversal: Many laws of physics describe real phenomena when the direction of time is reversed. Mathematically, this is represented by the transformation, ${\displaystyle t\,\rightarrow -t}$. For example, Newton's second law of motion still holds if, in the equation ${\displaystyle F\,=m{\ddot {r}}}$, ${\displaystyle t}$ is replaced by ${\displaystyle -t}$. This may be illustrated by recording the motion of an object thrown up vertically (neglecting air resistance) and then playing it back. The object will follow the same parabolic trajectory through the air, whether the recording is played normally or in reverse. Thus, position is symmetric with respect to the instant that the object is at its maximum height.
• Spatial inversion: These are represented by transformations of the form ${\displaystyle {\vec {r}}\,\rightarrow -{\vec {r}}}$ and indicate an invariance property of a system when the coordinates are 'inverted'. Stated another way, these are symmetries between a certain object and its mirror image.
• Glide reflection: These are represented by a composition of a translation and a reflection. These symmetries occur in some crystals and in some planar symmetries, known as wallpaper symmetries.

C, P, and T

Main article: Wu Experiment

The Standard Model of particle physics has three related natural near-symmetries. These state that the universe in which we live should be indistinguishable from one where a certain type of change is introduced.

• C-symmetry (charge symmetry), a universe where every particle is replaced with its antiparticle
• P-symmetry (parity symmetry), a universe where everything is mirrored along the three physical axes. This excludes weak interactions as demonstrated by Chien-Shiung Wu.

• T-symmetry (time reversal symmetry), a universe where the direction of time is reversed. T-symmetry is counterintuitive (the future and the past are not symmetrical) but explained by the fact that the Standard Model describes local properties, not global ones like entropy. To properly reverse the direction of time, one would have to put the Big Bang and the resulting low-entropy state in the "future". Since we perceive the "past" ("future") as having lower (higher) entropy than the present, the inhabitants of this hypothetical time-reversed universe would perceive the future in the same way as we perceive the past, and vice versa.

These symmetries are near-symmetries because each is broken in the present-day universe. However, the Standard Model predicts that the combination of the three (that is, the simultaneous application of all three transformations) must be a symmetry, called CPT symmetry. CP violation, the violation of the combination of C- and P-symmetry, is necessary for the presence of significant amounts of baryonic matter in the universe. CP violation is a fruitful area of current research in particle physics.

Supersymmetry

Main article: Supersymmetry

A type of symmetry known as supersymmetry has been used to try to make theoretical advances in the Standard Model. Supersymmetry is based on the idea that there is another physical symmetry beyond those already developed in the Standard Model, specifically a symmetry between bosons and fermions. Supersymmetry asserts that each type of boson has, as a supersymmetric partner, a fermion, called a superpartner, and vice versa. Supersymmetry has not yet been experimentally verified: no known particle has the correct properties to be a superpartner of any other known particle. Currently LHC is preparing for a run which tests supersymmetry.

Mathematics of physical symmetry

Main article: Symmetry group

See also: Symmetry in quantum mechanics and Symmetries in general relativity

The transformations describing physical symmetries typically form a mathematical group. Group theory is an important area of mathematics for physicists.

Continuous symmetries are specified mathematically by continuous groups (called Lie groups). Many physical symmetries are isometries and are specified by symmetry groups. Sometimes this term is used for more general types of symmetries. The set of all proper rotations (about any angle) through any axis of a sphere form a Lie group called the special orthogonal group SO(3). (The '3' refers to the three-dimensional space of an ordinary sphere.) Thus, the symmetry group of the sphere with proper rotations is SO(3). Any rotation preserves distances on the surface of the ball. The set of all Lorentz transformations form a group called the Lorentz group (this may be generalised to the Poincaré group).

Discrete groups describe discrete symmetries. For example, the symmetries of an equilateral triangle are characterized by the symmetric group S₃.

A type of physical theory based on local symmetries is called a gauge theory and the symmetries natural to such a theory are called gauge symmetries. Gauge symmetries in the Standard Model, used to describe three of the fundamental interactions, are based on the SU(3) × SU(2) × U(1) group. (Roughly speaking, the symmetries of the SU(3) group describe the strong force, the SU(2) group describes the weak interaction and the U(1) group describes the electromagnetic force.)

Also, the reduction by symmetry of the energy functional under the action by a group and spontaneous symmetry breaking of transformations of symmetric groups appear to elucidate topics in particle physics (for example, the unification of electromagnetism and the weak force in physical cosmology).

Conservation laws and symmetry

Main article: Noether's theorem

The symmetry properties of a physical system are intimately related to the conservation laws characterizing that system. Noether's theorem gives a precise description of this relation. The theorem states that each continuous symmetry of a physical system implies that some physical property of that system is conserved. Conversely, each conserved quantity has a corresponding symmetry. For example, spatial translation symmetry (i.e. homogeneity of space) gives rise to conservation of (linear) momentum, and temporal translation symmetry (i.e. homogeneity of time) gives rise to conservation of energy.

The following table summarizes some fundamental symmetries and the associated conserved quantity.

Class	Invariance	Conserved quantity
Proper orthochronous Lorentz symmetry	translation in time (homogeneity)	energy
	translation in space (homogeneity)	linear momentum
	rotation in space (isotropy)	angular momentum
	Lorentz-boost (isotropy)	mass moment N = tp − Er
Discrete symmetry	P, coordinate inversion	spatial parity
	C, charge conjugation	charge parity
	T, time reversal	time parity
	CPT	product of parities
Internal symmetry (independent of spacetime coordinates)	U(1) gauge transformation	electric charge
	U(1) gauge transformation	lepton generation number
	U(1) gauge transformation	hypercharge
	U(1)Y gauge transformation	weak hypercharge
	U(2) [ U(1) × SU(2) ]	electroweak force
	SU(2) gauge transformation	isospin
	SU(2)L gauge transformation	weak isospin
	P × SU(2)	G-parity
	SU(3) "winding number"	baryon number
	SU(3) gauge transformation	quark color
	SU(3) (approximate)	quark flavor
	S(U(2) × U(3)) [ U(1) × SU(2) × SU(3) ]	Standard Model

Mathematics

Continuous symmetries in physics preserve transformations. One can specify a symmetry by showing how a very small transformation affects various particle fields. The commutator of two of these infinitesimal transformations are equivalent to a third infinitesimal transformation of the same kind hence they form a Lie algebra.

A general coordinate transformation described as the general field ${\displaystyle h(x)}$ (also known as a diffeomorphism) has the infinitesimal effect on a scalar ${\displaystyle \phi (x)}$, spinor ${\displaystyle \psi (x)}$ or vector field ${\displaystyle A(x)}$ that can be expressed (using the Einstein summation convention):

${\displaystyle \delta \phi (x)=h^{\mu }(x)\partial _{\mu }\phi (x)}$

${\displaystyle \delta \psi ^{\alpha }(x)=h^{\mu }(x)\partial _{\mu }\psi ^{\alpha }(x)+\partial _{\mu }h_{\nu }(x)\sigma _{\mu \nu }^{\alpha \beta }\psi ^{\beta }(x)}$

${\displaystyle \delta A_{\mu }(x)=h^{\nu }(x)\partial _{\nu }A_{\mu }(x)+A_{\nu }(x)\partial _{\mu }h^{\nu }(x)}$

Without gravity only the Poincaré symmetries are preserved which restricts ${\displaystyle h(x)}$ to be of the form:

${\displaystyle h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }}$

where M is an antisymmetric matrix (giving the Lorentz and rotational symmetries) and P is a general vector (giving the translational symmetries). Other symmetries affect multiple fields simultaneously. For example, local gauge transformations apply to both a vector and spinor field:

${\displaystyle \delta \psi ^{\alpha }(x)=\lambda (x).\tau ^{\alpha \beta }\psi ^{\beta }(x)}$

${\displaystyle \delta A_{\mu }(x)=\partial _{\mu }\lambda (x),}$

where ${\displaystyle \tau }$ are generators of a particular Lie group. So far the transformations on the right have only included fields of the same type. Supersymmetries are defined according to how the mix fields of different types.

Another symmetry which is part of some theories of physics and not in others is scale invariance which involve Weyl transformations of the following kind:

${\displaystyle \delta \phi (x)=\Omega (x)\phi (x)}$

If the fields have this symmetry then it can be shown that the field theory is almost certainly conformally invariant also. This means that in the absence of gravity h(x) would restricted to the form:

${\displaystyle h^{\mu }(x)=M^{\mu \nu }x_{\nu }+P^{\mu }+Dx_{\mu }+K^{\mu }|x|^{2}-2K^{\nu }x_{\nu }x_{\mu },}$

with D generating scale transformations and K generating special conformal transformations. For example, N = 4 super-Yang–Mills theory has this symmetry while general relativity doesn't although other theories of gravity such as conformal gravity do. The 'action' of a field theory is an invariant under all the symmetries of the theory. Much of modern theoretical physics is to do with speculating on the various symmetries the Universe may have and finding the invariants to construct field theories as models.

In string theories, since a string can be decomposed into an infinite number of particle fields, the symmetries on the string world sheet is equivalent to special transformations which mix an infinite number of fields.

 

Yang–Mills theory

From Wikipedia, the free encyclopedia

https://en.wikipedia.org/wiki/Yang%E2%80%93Mills_theory

Yang–Mills theory in the non-perturbative regime: The equations of Yang–Mills remain unsolved at energy scales relevant for describing atomic nuclei. How does Yang–Mills theory give rise to the physics of nuclei and nuclear constituents?

Quantum field theory
Feynman diagram

In mathematical physics, Yang–Mills theory is a gauge theory based on a special unitary group SU(N), or more generally any compact, reductive Lie algebra. Yang–Mills theory seeks to describe the behavior of elementary particles using these non-abelian Lie groups and is at the core of the unification of the electromagnetic force and weak forces (i.e. U(1) × SU(2)) as well as quantum chromodynamics, the theory of the strong force (based on SU(3)). Thus it forms the basis of our understanding of the Standard Model of particle physics.

History and theoretical description

In 1953, in a private correspondence, Wolfgang Pauli formulated a six-dimensional theory of Einstein's field equations of general relativity, extending the five-dimensional theory of Kaluza, Klein, Fock and others to a higher-dimensional internal space. However, there is no evidence that Pauli developed the Lagrangian of a gauge field or the quantization of it. Because Pauli found that his theory "leads to some rather unphysical shadow particles", he refrained from publishing his results formally. Although Pauli did not publish his six-dimensional theory, he gave two talks about it in Zürich. Recent research shows that an extended Kaluza–Klein theory is in general not equivalent to Yang–Mills theory, as the former contains additional terms. Chen Ning Yang long considered the idea of non-abelian gauge theories. Only after meeting Robert Mills did he introduce the junior scientist to the idea and lay the key hypothesis that Mills would use to assist in creating a new theory. This eventually became the Yang-Mills Theory, as Mills himself discussed,

"During the academic year 1953-1954, Yang was a visitor to Brookhaven National Laboratory...I was at Brookhaven also...and was assigned to the same office as Yang. Yang, who has demonstrated on a number of occasions his generosity to physicists beginning their careers, told me about his idea of generalizing gauge invariance and we discussed it at some length...I was able to contribute something to the discussions, especially with regard to the quantization procedures, and to a small degree in working out the formalism; however, the key ideas were Yang's."

In early 1954, Yang and Mills extended the concept of gauge theory for abelian groups, e.g. quantum electrodynamics, to non-abelian groups to provide an explanation for strong interactions. The idea by Yang–Mills was criticized by Pauli, as the quanta of the Yang–Mills field must be massless in order to maintain gauge invariance. The idea was set aside until 1960, when the concept of particles acquiring mass through symmetry breaking in massless theories was put forward, initially by Jeffrey Goldstone, Yoichiro Nambu, and Giovanni Jona-Lasinio.

This prompted a significant restart of Yang–Mills theory studies that proved successful in the formulation of both electroweak unification and quantum chromodynamics (QCD). The electroweak interaction is described by the gauge group SU(2) × U(1), while QCD is an SU(3) Yang–Mills theory. The massless gauge bosons of the electroweak SU(2) × U(1) mix after spontaneous symmetry breaking to produce the 3 massive weak bosons (
W⁺
,
W⁻
, and
Z
) as well as the still-massless photon field. The dynamics of the photon field and its interactions with matter are, in turn, governed by the U(1) gauge theory of quantum electrodynamics. The Standard Model combines the strong interaction with the unified electroweak interaction (unifying the weak and electromagnetic interaction) through the symmetry group SU(3) × SU(2) × U(1). In the current epoch the strong interaction is not unified with the electroweak interaction, but from the observed running of the coupling constants it is believed they all converge to a single value at very high energies.

Phenomenology at lower energies in quantum chromodynamics is not completely understood due to the difficulties of managing such a theory with a strong coupling. This may be the reason why confinement has not been theoretically proven, though it is a consistent experimental observation. This shows why QCD confinement at low energy is a mathematical problem of great relevance, and why the Yang–Mills existence and mass gap problem is a Millennium Prize Problem.

Mathematical overview

See also: Yang–Mills equations

The dx¹⊗σ₃ coefficient of a BPST instanton on the (x¹,x²)-slice of R⁴ where σ₃ is the third Pauli matrix (top left). The dx²⊗σ₃ coefficient (top right). These coefficients determine the restriction of the BPST instanton A with g=2,ρ=1,z=0 to this slice. The corresponding field strength centered around z=0 (bottom left). A visual representation of the field strength of a BPST instanton with center z on the compactification S⁴ of R⁴ (bottom right). The BPST instanton is a classical instanton solution to the Yang–Mills equations on R⁴.

Yang–Mills theories are special examples of gauge theories with a non-abelian symmetry group given by the Lagrangian

${\displaystyle {\mathcal {L}}_{\mathrm {gf} }=-{\frac {1}{2}}\operatorname {Tr} (F^{2})=-{\frac {1}{4}}F^{a\mu \nu }F_{\mu \nu }^{a}}$

with the generators ${\displaystyle T^{a}}$ of the Lie algebra, indexed by a, corresponding to the F-quantities (the curvature or field-strength form) satisfying

${\displaystyle \operatorname {Tr} (T^{a}T^{b})={\frac {1}{2}}\delta ^{ab},\quad [T^{a},T^{b}]=if^{abc}T^{c}.}$

Here, the f ^abc are structure constants of the Lie algebra (totally antisymmetric if the generators of the Lie algebra are normalised such that ${\displaystyle Tr(T^{a}T^{b})}$ is proportional with ${\displaystyle \delta ^{ab}}$), the covariant derivative is defined as

${\displaystyle D_{\mu }=I\partial _{\mu }-igT^{a}A_{\mu }^{a},}$

I is the identity matrix (matching the size of the generators), ${\displaystyle A_{\mu }^{a}}$ is the vector potential, and g is the coupling constant. In four dimensions, the coupling constant g is a pure number and for a SU(N) group one has ${\displaystyle a,b,c=1\ldots N^{2}-1.}$

The relation

${\displaystyle F_{\mu \nu }^{a}=\partial _{\mu }A_{\nu }^{a}-\partial _{\nu }A_{\mu }^{a}+gf^{abc}A_{\mu }^{b}A_{\nu }^{c}}$

can be derived by the commutator

${\displaystyle [D_{\mu },D_{\nu }]=-igT^{a}F_{\mu \nu }^{a}.}$

The field has the property of being self-interacting and the equations of motion that one obtains are said to be semilinear, as nonlinearities are both with and without derivatives. This means that one can manage this theory only by perturbation theory with small nonlinearities.

Note that the transition between "upper" ("contravariant") and "lower" ("covariant") vector or tensor components is trivial for a indices (e.g. ${\displaystyle f^{abc}=f_{abc}}$), whereas for μ and ν it is nontrivial, corresponding e.g. to the usual Lorentz signature, ${\displaystyle \eta _{\mu \nu }={\rm {diag}}(+---)}$.

From the given Lagrangian one can derive the equations of motion given by

${\displaystyle \partial ^{\mu }F_{\mu \nu }^{a}+gf^{abc}A^{\mu b}F_{\mu \nu }^{c}=0.}$

Putting ${\displaystyle F_{\mu \nu }=T^{a}F_{\mu \nu }^{a}}$, these can be rewritten as

${\displaystyle (D^{\mu }F_{\mu \nu })^{a}=0.}$

A Bianchi identity holds

${\displaystyle (D_{\mu }F_{\nu \kappa })^{a}+(D_{\kappa }F_{\mu \nu })^{a}+(D_{\nu }F_{\kappa \mu })^{a}=0}$

which is equivalent to the Jacobi identity

${\displaystyle [D_{\mu },[D_{\nu },D_{\kappa }]]+[D_{\kappa },[D_{\mu },D_{\nu }]]+[D_{\nu },[D_{\kappa },D_{\mu }]]=0}$

since ${\displaystyle [D_{\mu },F_{\nu \kappa }^{a}]=D_{\mu }F_{\nu \kappa }^{a}}$. Define the dual strength tensor ${\displaystyle {\tilde {F}}^{\mu \nu }={\frac {1}{2}}\varepsilon ^{\mu \nu \rho \sigma }F_{\rho \sigma }}$, then the Bianchi identity can be rewritten as

${\displaystyle D_{\mu }{\tilde {F}}^{\mu \nu }=0.}$

A source ${\displaystyle J_{\mu }^{a}}$ enters into the equations of motion as

${\displaystyle \partial ^{\mu }F_{\mu \nu }^{a}+gf^{abc}A^{b\mu }F_{\mu \nu }^{c}=-J_{\nu }^{a}.}$

Note that the currents must properly change under gauge group transformations.

We give here some comments about the physical dimensions of the coupling. In D dimensions, the field scales as ${\displaystyle [A]=[L^{\frac {2-D}{2}}]}$ and so the coupling must scale as ${\displaystyle [g^{2}]=[L^{D-4}]}$. This implies that Yang–Mills theory is not renormalizable for dimensions greater than four. Furthermore, for D = 4, the coupling is dimensionless and both the field and the square of the coupling have the same dimensions of the field and the coupling of a massless quartic scalar field theory. So, these theories share the scale invariance at the classical level.

Quantization

A method of quantizing the Yang–Mills theory is by functional methods, i.e. path integrals. One introduces a generating functional for n-point functions as

${\displaystyle Z[j]=\int [dA]\exp \left[-{\frac {i}{2}}\int d^{4}x\operatorname {Tr} (F^{\mu \nu }F_{\mu \nu })+i\int d^{4}x\,j_{\mu }^{a}(x)A^{a\mu }(x)\right],}$

but this integral has no meaning as it is because the potential vector can be arbitrarily chosen due to the gauge freedom. This problem was already known for quantum electrodynamics but here becomes more severe due to non-abelian properties of the gauge group. A way out has been given by Ludvig Faddeev and Victor Popov with the introduction of a ghost field (see Faddeev–Popov ghost) that has the property of being unphysical since, although it agrees with Fermi–Dirac statistics, it is a complex scalar field, which violates the spin–statistics theorem. So, we can write the generating functional as

${\displaystyle {\begin{aligned}Z[j,{\bar {\varepsilon }},\varepsilon ]&=\int [dA][d{\bar {c}}][dc]\exp \left\{iS_{F}[\partial A,A]+iS_{gf}[\partial A]+iS_{g}[\partial c,\partial {\bar {c}},c,{\bar {c}},A]\right\}\\&\exp \left\{i\int d^{4}xj_{\mu }^{a}(x)A^{a\mu }(x)+i\int d^{4}x[{\bar {c}}^{a}(x)\varepsilon ^{a}(x)+{\bar {\varepsilon }}^{a}(x)c^{a}(x)]\right\}\end{aligned}}}$

being

${\displaystyle S_{F}=-{\frac {1}{2}}\int \operatorname {d} \!^{4}x\operatorname {Tr} (F^{\mu \nu }F_{\mu \nu })}$

for the field,

${\displaystyle S_{gf}=-{\frac {1}{2\xi }}\int \operatorname {d} \!^{4}x(\partial \cdot A)^{2}}$

for the gauge fixing and

${\displaystyle S_{g}=-\int \operatorname {d} \!^{4}x({\bar {c}}^{a}\partial _{\mu }\partial ^{\mu }c^{a}+g{\bar {c}}^{a}f^{abc}\partial _{\mu }A^{b\mu }c^{c})}$

for the ghost. This is the expression commonly used to derive Feynman's rules (see Feynman diagram). Here we have c^a for the ghost field while ξ fixes the gauge's choice for the quantization. Feynman's rules obtained from this functional are the following

These rules for Feynman diagrams can be obtained when the generating functional given above is rewritten as

${\displaystyle {\begin{aligned}Z[j,{\bar {\varepsilon }},\varepsilon ]&=\exp \left(-ig\int d^{4}x\,{\frac {\delta }{i\delta {\bar {\varepsilon }}^{a}(x)}}f^{abc}\partial _{\mu }{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta \varepsilon ^{c}(x)}}\right)\\&\qquad \times \exp \left(-ig\int d^{4}xf^{abc}\partial _{\mu }{\frac {i\delta }{\delta j_{\nu }^{a}(x)}}{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta j^{c\nu }(x)}}\right)\\&\qquad \qquad \times \exp \left(-i{\frac {g^{2}}{4}}\int d^{4}xf^{abc}f^{ars}{\frac {i\delta }{\delta j_{\mu }^{b}(x)}}{\frac {i\delta }{\delta j_{\nu }^{c}(x)}}{\frac {i\delta }{\delta j^{r\mu }(x)}}{\frac {i\delta }{\delta j^{s\nu }(x)}}\right)\\&\qquad \qquad \qquad \times Z_{0}[j,{\bar {\varepsilon }},\varepsilon ]\end{aligned}}}$

with

${\displaystyle Z_{0}[j,{\bar {\varepsilon }},\varepsilon ]=\exp \left(-\int d^{4}xd^{4}y{\bar {\varepsilon }}^{a}(x)C^{ab}(x-y)\varepsilon ^{b}(y)\right)\exp \left({\tfrac {1}{2}}\int d^{4}xd^{4}yj_{\mu }^{a}(x)D^{ab\mu \nu }(x-y)j_{\nu }^{b}(y)\right)}$

being the generating functional of the free theory. Expanding in g and computing the functional derivatives, we are able to obtain all the n-point functions with perturbation theory. Using LSZ reduction formula we get from the n-point functions the corresponding process amplitudes, cross sections and decay rates. The theory is renormalizable and corrections are finite at any order of perturbation theory.

For quantum electrodynamics the ghost field decouples because the gauge group is abelian. This can be seen from the coupling between the gauge field and the ghost field that is ${\displaystyle {\bar {c}}^{a}f^{abc}\partial _{\mu }A^{b\mu }c^{c}}$. For the abelian case, all the structure constants ${\displaystyle f^{abc}}$ are zero and so there is no coupling. In the non-abelian case, the ghost field appears as a useful way to rewrite the quantum field theory without physical consequences on the observables of the theory such as cross sections or decay rates.

One of the most important results obtained for Yang–Mills theory is asymptotic freedom. This result can be obtained by assuming that the coupling constant g is small (so small nonlinearities), as for high energies, and applying perturbation theory. The relevance of this result is due to the fact that a Yang–Mills theory that describes strong interaction and asymptotic freedom permits proper treatment of experimental results coming from deep inelastic scattering.

To obtain the behavior of the Yang–Mills theory at high energies, and so to prove asymptotic freedom, one applies perturbation theory assuming a small coupling. This is verified a posteriori in the ultraviolet limit. In the opposite limit, the infrared limit, the situation is the opposite, as the coupling is too large for perturbation theory to be reliable. Most of the difficulties that research meets is just managing the theory at low energies. That is the interesting case, being inherent to the description of hadronic matter and, more generally, to all the observed bound states of gluons and quarks and their confinement (see hadrons). The most used method to study the theory in this limit is to try to solve it on computers (see lattice gauge theory). In this case, large computational resources are needed to be sure the correct limit of infinite volume (smaller lattice spacing) is obtained. This is the limit the results must be compared with. Smaller spacing and larger coupling are not independent of each other, and larger computational resources are needed for each. As of today, the situation appears somewhat satisfactory for the hadronic spectrum and the computation of the gluon and ghost propagators, but the glueball and hybrids spectra are yet a questioned matter in view of the experimental observation of such exotic states. Indeed, the σ resonance is not seen in any of such lattice computations and contrasting interpretations have been put forward. This is a hotly debated issue.

Open problems

Yang–Mills theories met with general acceptance in the physics community after Gerard 't Hooft, in 1972, worked out their renormalization, relying on a formulation of the problem worked out by his advisor Martinus Veltman. Renormalizability is obtained even if the gauge bosons described by this theory are massive, as in the electroweak theory, provided the mass is only an "acquired" one, generated by the Higgs mechanism.

The mathematics of the Yang–Mills theory is a very active field of research, yielding e.g. invariants of differentiable structures on four-dimensional manifolds via work of Simon Donaldson. Furthermore, the field of Yang–Mills theories was included in the Clay Mathematics Institute's list of "Millennium Prize Problems". Here the prize-problem consists, especially, in a proof of the conjecture that the lowest excitations of a pure Yang–Mills theory (i.e. without matter fields) have a finite mass-gap with regard to the vacuum state. Another open problem, connected with this conjecture, is a proof of the confinement property in the presence of additional Fermion particles.

In physics the survey of Yang–Mills theories does not usually start from perturbation analysis or analytical methods, but more recently from systematic application of numerical methods to lattice gauge theories.